An air-conditioner housing comprising an evaporation device, a heating device, and a mixing chamber, out of which air flows to the rear footwell while serving to ventilate the rear passenger compartment. The air-conditioner housing also comprises an air control device that uses mixing flaps to control the air flowing via the evaporation device and the heating device into the mixing chamber. The air-conditioner housing has a separate stratification duct through which cool air can be guided into the ventilation area of the rear passenger compartment.